The Fermi ? Ulam problem deals with the complex dynamics of a classical point particle bouncing between harmonically oscillating walls (a dynamical billiard) [1,2]. Though the problem can be applied to various fields of physics, similar problems with respect to solitons were not considered yet, as far as we know. In the talk, we analyze features of ?longitudinal? and ?transverse? solitons in a dynamical billiard. For definiteness, we consider the atomic Bose ? Einstein condensate (BEC) in a dynamical trap with oscillating walls [3-5].

The initial governing equation is the Gross ? Pitaevskii equation for the wave function of weakly non-ideal atomic gas at zero temperature. For BEC confined by the trap in the transverse directions and ideal barriers with square potential moving along the longitudinal direction, the BEC wavefunction obeys the nonlinear Schr?dinger equation with zero conditions at oscillating boundaries. In unbounded scheme (without barriers) the equation is solvable by the inverse scattering method and support sech-type solitons [6]. Such ?longitudinal? soliton interaction with an ideal motionless barrier can be described as interaction of this soliton with its antiphased mirror image. For harmonically oscillating barriers, we demonstrate various scenarios of the interaction including soliton quasi-periodic and chaotic motion.

In regimes when the BEC wave packet diffuses over the whole trap length, the governing equation for the case of harmonically oscillating barriers has highly non-equidistant quasienergy spectrum. If the oscillation frequency is close to the frequency of transition between a pair of the levels, it is possible to realize their resonance interaction in the framework of two-level scheme. For these levels? amplitudes the transverse dynamics is governing by a Manakov-type system of equations with coherent linear and incoherent nonlinear coupling. We present families of ?transverse? two-component (vector) solitons corresponding to these equations and various scenarios of solitons? interaction including breather formation.

[1] E. Fermi. ?On the origin of the cosmic radiation,? Phys. Rev. 75, 1169 ? 1174 (1949).
[2] S. Ulam. ?On some statistical properties of dynamical systems,? in Proceedings of the Fourth Berkley Symposium on Mathematics, Statistics, and Probability (California University Press, Berkeley ? Los Angeles, 1961). Vol. 3, pp. 315-320.
[3] N. N. Rosanov, ?Nonlinear Rabi oscillations in a Bose ? Einstein condensate,? Phys. Rev. A 88, 063616 (2013).
[4] N. N. Rozanov and G. B. Sochilin, ?Quasi-energy of single quantum particles and a Bose?Einstein condensate in a dynamical trap,? Journal of Experimental and Theoretical Physics 118, 124?132 (2014).
[5] N. N. Rosanov, ?Vector solitons of a Bose ? Einstein condensate in a dynamical trap,? Phys. Rev. A 89, 035601 (2014).
[6] V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of solitons: The inverse scattering transform. Moscow: Nauka, 1980 (in Russian); English translation: Consultant Bureau, New York 1984.